b parameter of renormalized beta distribution

Average Error: 0.1 → 0.2

Time: 32.3s

Precision: binary64

Cost: 964

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{m}{v} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{2 - \left(m + 1\right)}{v}\\ \end{array} \]

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.8e-17)
   (- (/ m v) 1.0)
   (* (* m (- 1.0 m)) (/ (- 2.0 (+ m 1.0)) v))))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

↓

double code(double m, double v) {
	double tmp;
	if (m <= 1.8e-17) {
		tmp = (m / v) - 1.0;
	} else {
		tmp = (m * (1.0 - m)) * ((2.0 - (m + 1.0)) / v);
	}
	return tmp;
}

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * (1.0d0 - m)
end function

↓

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    real(8) :: tmp
    if (m <= 1.8d-17) then
        tmp = (m / v) - 1.0d0
    else
        tmp = (m * (1.0d0 - m)) * ((2.0d0 - (m + 1.0d0)) / v)
    end if
    code = tmp
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

↓

public static double code(double m, double v) {
	double tmp;
	if (m <= 1.8e-17) {
		tmp = (m / v) - 1.0;
	} else {
		tmp = (m * (1.0 - m)) * ((2.0 - (m + 1.0)) / v);
	}
	return tmp;
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m)

↓

def code(m, v):
	tmp = 0
	if m <= 1.8e-17:
		tmp = (m / v) - 1.0
	else:
		tmp = (m * (1.0 - m)) * ((2.0 - (m + 1.0)) / v)
	return tmp

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * Float64(1.0 - m))
end

↓

function code(m, v)
	tmp = 0.0
	if (m <= 1.8e-17)
		tmp = Float64(Float64(m / v) - 1.0);
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) * Float64(Float64(2.0 - Float64(m + 1.0)) / v));
	end
	return tmp
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
end

↓

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 1.8e-17)
		tmp = (m / v) - 1.0;
	else
		tmp = (m * (1.0 - m)) * ((2.0 - (m + 1.0)) / v);
	end
	tmp_2 = tmp;
end

Wolfram

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

↓

code[m_, v_] := If[LessEqual[m, 1.8e-17], N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 - N[(m + 1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.79999999999999997e-17

   1. Initial program 0.0

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]

   2. Taylor expanded in m around 0 0.1

      \[\leadsto \color{blue}{\left(1 + \frac{1}{v}\right) \cdot m - 1} \]

   3. Taylor expanded in v around 0 0.0

      \[\leadsto \color{blue}{\frac{m}{v}} - 1 \]

   if 1.79999999999999997e-17 < m

   1. Initial program 0.4

      \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]

   2. Taylor expanded in v around 0 1.0

      \[\leadsto \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \cdot \left(1 - m\right) \]

   3. Applied egg-rr 1.0

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v} + 0} \]

   4. Simplified 1.0

      \[\leadsto \color{blue}{\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}} \]
      Proof

      [Start] 1.0	\[ \left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v} + 0 \]
      rational_best-simplify-3 [=>] 1.0	\[ \color{blue}{0 + \left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v}} \]
      rational_best-simplify-6 [=>] 1.0	\[ \color{blue}{\left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v}} \]
      rational_best-simplify-55 [=>] 1.0	\[ \color{blue}{\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}} \]

   5. Taylor expanded in m around 0 1.0

      \[\leadsto \left(m \cdot \left(1 - m\right)\right) \cdot \color{blue}{\left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right)} \]

   6. Simplified 1.0

      \[\leadsto \left(m \cdot \left(1 - m\right)\right) \cdot \color{blue}{\frac{2 - \left(m + 1\right)}{v}} \]
      Proof

      [Start] 1.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \left(-1 \cdot \frac{m}{v} + \frac{1}{v}\right) \]
      rational_best-simplify-3 [=>] 1.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \color{blue}{\left(\frac{1}{v} + -1 \cdot \frac{m}{v}\right)} \]
      metadata-eval [<=] 1.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \left(\frac{\color{blue}{2 - 1}}{v} + -1 \cdot \frac{m}{v}\right) \]
      rational_best-simplify-66 [<=] 1.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \left(\color{blue}{\left(\frac{2}{v} - \frac{1}{v}\right)} + -1 \cdot \frac{m}{v}\right) \]
      rational_best-simplify-1 [=>] 1.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \left(\left(\frac{2}{v} - \frac{1}{v}\right) + \color{blue}{\frac{m}{v} \cdot -1}\right) \]
      rational_best-simplify-10 [=>] 1.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \left(\left(\frac{2}{v} - \frac{1}{v}\right) + \color{blue}{\left(-\frac{m}{v}\right)}\right) \]
      rational_best-simplify-56 [=>] 1.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \color{blue}{\left(\frac{2}{v} - \left(\frac{1}{v} + \frac{m}{v}\right)\right)} \]
      rational_best-simplify-64 [=>] 1.1	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \left(\frac{2}{v} - \color{blue}{\frac{1 + m}{v}}\right) \]
      rational_best-simplify-66 [=>] 1.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \color{blue}{\frac{2 - \left(1 + m\right)}{v}} \]
      rational_best-simplify-3 [=>] 1.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \frac{2 - \color{blue}{\left(m + 1\right)}}{v} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{m}{v} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{2 - \left(m + 1\right)}{v}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	1088

\[\left(\frac{m \cdot \left(m + \left(m + -2\right)\right)}{v \cdot -2} - 1\right) \cdot \left(1 - m\right) \]

Alternative 2
Error	25.1
Cost	852

\[\begin{array}{l} \mathbf{if}\;v \leq 1.7 \cdot 10^{-210}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;v \leq 2.05 \cdot 10^{-199}:\\ \;\;\;\;-1\\ \mathbf{elif}\;v \leq 1.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-129}:\\ \;\;\;\;-1\\ \mathbf{elif}\;v \leq 3.8 \cdot 10^{-122}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	0.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{m}{v} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	832

\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]

Alternative 5
Error	2.3
Cost	772

\[\begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{-v}\\ \end{array} \]

Alternative 6
Error	24.5
Cost	652

\[\begin{array}{l} t_0 := -\left(1 - m\right)\\ \mathbf{if}\;v \leq 1.65 \cdot 10^{-210}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;v \leq 1.7 \cdot 10^{-199}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 1.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	9.4
Cost	448

\[\left(\frac{m}{v} + m\right) - 1 \]

Alternative 8
Error	9.4
Cost	320

\[\frac{m}{v} - 1 \]

Alternative 9
Error	36.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))